Prove: If H is a finite group of even cardinality, the H contains an element h of order 2.
Problem: Suppose is even. Prove that at least one element of is of order two.
Proof: Suppose that contained no elements of order two. We can see then that for all nontrivial elements of . Therefore elements of this nature come in distinct pairs ( ) adding up all these elements will then give an even number. Then noting that since no nontrivial element of had order two we can see that the total number of elements in is the number of all elements whose order isn't two and the identity element. But this is an even number plus one, thus odd which is of course a contradiction.